Question: What is the least dense rigid periodic circle packing of uniform coordination?
In other words: Using congruent circles, how sparse can a rigid and periodic packing (where each circle touches the same number of other circles) be made?
Here is some clarification on the terminology:
By "congruent" I mean identical.
By "how sparse" I mean the rarest possible circle packing.
By "rigid" I mean a packing in which no circle can be moved without moving other circles. Maybe the word stable would have been better. Sorry about that.
By "uniform coordination" I mean a packing where each circle is in contact with $n$ other circles.
The packing has to be finite since I asked for a periodic packing.